applied sciences Article
Member Discrete Element Method for Static and Dynamic Responses Analysis of Steel Frames with Semi-Rigid Joints Jihong Ye 1, * and Lingling Xu 2 1 2
*
State Key Laboratory for Geomechanics & Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China Key Laboratory of Concrete and Pre-Stressed Concrete Structures of the Ministry of Education, Southeast University, Nanjing 210018, China;
[email protected] Correspondence:
[email protected]; Tel.: +86-516-8359-0609
Academic Editor: Zheng Lu Received: 27 May 2017; Accepted: 7 July 2017; Published: 11 July 2017
Featured Application: The MDEM can naturally deal with strong nonlinearity and discontinuity. A unified computational framework is applied for static and dynamic analyses. The method is an effective tool to investigate complex behaviors of steel frames with semi-rigid connections. Abstract: In this paper, a simple and effective numerical approach is presented on the basis of the Member Discrete Element Method (MDEM) to investigate static and dynamic responses of steel frames with semi-rigid joints. In the MDEM, structures are discretized into a set of finite rigid particles. The motion equation of each particle is solved by the central difference method and two adjacent arbitrarily particles are connected by the contact constitutive model. The above characteristics means that the MDEM is able to naturally handle structural geometric nonlinearity and fracture. Meanwhile, the computational framework of static analysis is consistent with that of dynamic analysis, except the determination of damping. A virtual spring element with two particles but without actual mass and length is used to simulate the mechanical behaviors of semi-rigid joints. The spring element is not directly involved in the calculation, but is employed only to modify the stiffness coefficients of contact elements at the semi-rigid connections. Based on the above-mentioned concept, the modified formula of the contact element stiffness with consideration of semi-rigid connections is deduced. The Richard-Abbort four-parameter model and independent hardening model are further introduced accordingly to accurately capture the nonlinearity and hysteresis performance of semi-rigid connections. Finally, the numerical approach proposed is verified by complex behaviors of steel frames with semi-rigid connections such as geometric nonlinearity, snap-through buckling, dynamic responses and fracture. The comparison of static and dynamic responses obtained using the modified MDEM and those of the published studies illustrates that the modified MDEM can simulate the mechanical behaviors of semi-rigid connections simply and directly, and can accurately effectively capture the linear and nonlinear behaviors of semi-rigid connections under static and dynamic loading. Some conclusions, as expected, are drawn that structural bearing capacity under static loading will be overestimated if semi-rigid connections are ignored; when the frequency of dynamic load applied is close to structural fundamental frequency, hysteresis damping of nonlinear semi-rigid connections can cause energy dissipation compared to rigid and linear semi-rigid connections, thus avoiding the occurrence of resonance. Additionally, fracture analysis also indicates that semi-rigid steel frames possess more anti-collapse capacity than that with rigid steel frames. Keywords: steel frames with semi-rigid joints; Member Discrete Element Method; nonlinear semi-rigid connection; geometric nonlinearity; snap-through buckling; dynamic response; fracture
Appl. Sci. 2017, 7, 714; doi:10.3390/app7070714
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1. Introduction Beam-to-column joints are frequently simplified to be fully rigid or pinned connections in structural analyses and designs, while real beam-to-column joints are semi-rigid connections between the two extreme cases. Compared with rigid connections, semi-rigid connections will reduce total stiffness of steel frames and make lateral displacement increase, thereby causing a second-order effect [1]. Therefore, the rigidity of a semi-rigid joint has a significant effect on the strength and displacement response of steel frames. At present, approaches for investigating mechanical behaviors of semi-rigid joints fall into three categories: experiment, mathematical model and numerical simulation. For the first two approaches, an amount of mature research has been carried out, and a variety of linear or nonlinear semi-rigid connection models were proposed [2–6]. Furthermore, Eurccode-3 (EC3) [7] provides the M-θ curves for some specified beam-to column joints of steel structures. Based on the M-θ curves of EC3, Keulen et al. [8] introduced the half initial secant stiffness approach to simplify the full characteristics of nonlinear connection stiffness curve into a bilinear curve in practical design, and analysis results of the two cases were in good agreement. Thus, numerical simulation methods of semi-rigid joints have increasingly attracted the attentions of numerous researchers in recent years. The finite element method (FEM) is the most commonly used method in the numerical simulation of semi-rigid joints. Lui and Chen [9], Ho and Chen [10] considered structural geometric nonlinearity by stability functions, and analyzed nonlinearity and buckling behaviors of semi-rigid beam-to-column joints in conjunction with the geometric stiffness matrix and updated Lagrangian approach. Yau and Chan [11] presented a beam-column element with springs connected in series for geometric and material nonlinear analysis of steel frames with semi-rigid connections, as well as proposed an efficient approach to trace the equilibrium path of steel frames. Zhou and Chan [12] performed second-order analysis of steel frames with semi-rigid connections by a displacement-based finite-element approach. Chui and Chan [13,14], Sophianopoulos [15] investigated the dynamic responses of steel frames with semi-rigid joints, where a semi-rigid joint was simulated by a zero-length spring element. Li et al. [16] and Rodrigues et al. [17] presented a multi-degree of freedom spring system to represent physical models of semi-rigid connections and took semi-rigid connections as new elements. This system transferred axial forces, shearing forces as well as bending moments. All the above results are based on the conventional finite-element method. However, in the static or dynamic analyses especially for strong nonlinear and discontinuous problems, the method has some limits such as difficult modeling, plenty of unknowns, long consuming time and convergence difficulty. In order to shorten consuming time, Nguyen and Kim [18,19] proposed a numerical procedure based on the beam-column method, and performed nonlinearly elastic and inelastic time-history analysis of three-dimensional semi-rigid steel frames by using the based-displacement finite element method. Compared to SAP2000(Computers and Structures, Inc., California, CA, USA), the procedure captured the second-order effect by using only one element per member, and reduced computational time. Nevertheless, it failed to model fracture behaviors of steel frames with semi-rigid joints. Lien and Chiou [20], Yu and Zhu [21] carried out an investigation on dynamic collapse of semi-rigid steel frames using vector formed intrinsic finite-element (VFIFE) method. A zero-length spring element was employed to simulate the semi-rigid joints. However, the spring element is involved in the calculation, which makes that the method has a cumbersome calculation process and long consuming time. In addition, for discontinuous issues, this method needs to regenerate a new particle at the point where fracture occurs, causing the computation time to increase again. Different from the aforementioned methods derived from continuum mechanics, the theoretical basis of Member Discrete Element Method (MDEM) presented by Ye team [22] according to conventional Particle Discrete Element [23] is non-continuum mechanics. In the MDEM, a member is represented by a set of finite particle rather than by continuum. The motion equation of each particle is solved by the central difference method and arbitrarily adjacent particles are connected by the contact constitutive model. The difference between the MDEM and conventional DEM lies in the
basis of Member Discrete Element Method (MDEM) presented by Ye team [22] according to conventional Particle Discrete Element [23] is non-continuum mechanics. In the MDEM, a member is represented by a set of finite particle rather than by continuum. The motion equation of each particle is solved by the central difference method and arbitrarily adjacent particles are connected by the contact constitutive model. The difference between the MDEM and conventional DEM lies in Appl. Sci. 2017, 7, 714 3 ofthe 23 establishment of the contact constitutive model. Therefore, the MDEM not only extends the application range of conventional DEM, but also inherits its advantages such as no assembly of establishment of the constitutive Therefore, theno MDEM only extends the application stiffness matrix andcontact iterative solution ofmodel. motion equations, specialnot treatment for nonlinear issues, range of conventional DEM, but also inherits its advantages such as no assembly of stiffness matrix no re-meshing for fracture problems, and no convergence difficulty. At present, the method has been and iterative solution equations, no special treatment for nonlinear no [25] re-meshing for successfully applied of to motion geometric nonlinearity [24] and nonlinear dynamicissues, analysis of member fracture problems, and no convergence difficulty. At present, the method has been successfully applied structures as well as progressive collapse simulation of reticulated shells [22]. These results show that to geometric [24] and nonlinear analysis [25] of member structures asmaterial well as the MDEM nonlinearity is a simple and effective tool dynamic to process geometrically large deformation, progressive simulation of However, reticulatedno shells [22].that These that thefor MDEM a nonlinearitycollapse and fracture issues. report the results MDEMshow is adopted static is and simple and effective tool to process geometrically large deformation, material nonlinearity and fracture dynamic response analysis of steel frames with semi-rigid joints exists. issues.AHowever, no report thatnumerical the MDEMapproach is adopted static and dynamic response analysis of steel simple and effective is for proposed based on the MDEM for static and frames with semi-rigid joints exists. dynamic response analysis of steel frames with semi-rigid joints. Brief details of the basic theory for A simple and effective numerical approach is proposed based on the MDEM for static andgeometric dynamic the MDEM are first presented including particle motion equations, particle internal forces, response analysis of steeland frames with semi-rigid Brief details of the basic theory for by theaMDEM nonlinearity modelling fracture modeling. joints. The semi-rigid connection is simulated virtual are first presented including particle motion equations, particle internal forces, geometric nonlinearity spring element, and then the modified formula of the contact element stiffness adjacent to the semimodelling and fracture modeling. The semi-rigid connection is simulated by a virtual spring element, rigid connection is derived. The nonlinear behaviors of semi-rigid connections are captured by the and then semi-rigid the modified formula ofmodel the contact element stiffnessfour-parameter adjacent to the semi-rigid connection its is bilinear connection and Richard-Abbort model. Furthermore, derived. The nonlinear behaviors of semi-rigid connections are captured by the bilinear semi-rigid hysteresis behavior is traced by the independent hardening model. The correctness and accuracy of connection model and Richard-Abbort model.behaviors Furthermore, its hysteresis behavior is the numerical procedure proposed arefour-parameter verified by complex analyses of steel frames with traced by the independent hardening model. The correctness and accuracy of the numerical procedure semi-rigid connections, such as geometric nonlinearity, snap-through buckling, dynamic responses proposed are verified by complex behaviors analyses of steel frames with semi-rigid connections, such and fracture. as geometric nonlinearity, snap-through buckling, dynamic responses and fracture. 2. Member Discrete Element Method (MDEM) 2. Member Discrete Element Method (MDEM) 2.1. Particle Motion Equations and Internal Forces 2.1. Particle Motion Equations and Internal Forces The Member Discrete Element Method (MDEM) presented by Ye team [22] discretizes a steel The Member Discrete Element Method (MDEM) presented by Ye team [22] discretizes a steel frame into a set of finite rigid particles, and adjacent particles (e.g., Particles A and B) are connected frame into a set of finite rigid particles, and adjacent particles (e.g., Particles A and B) are connected by the contact element (i.e., contact constitutive model), as shown in Figure 1. The criterion for the by the contact element (i.e., contact constitutive model), as shown in Figure 1. The criterion for the number of particles in the MDEM is consistent with that for beam elements in FEM, which is detailed number of particles in the MDEM is consistent with that for beam elements in FEM, which is detailed in Reference [26]. The method not only overcomes the difficulties of conventional FEM in processing in Reference [26]. The method not only overcomes the difficulties of conventional FEM in processing strong nonlinearity and discontinuity, but also eliminates the shortcomings with low precision, long strong nonlinearity and discontinuity, but also eliminates the shortcomings with low precision, long consuming time and poor applicability of conventional DEM. The calculation framework of the consuming time and poor applicability of conventional DEM. The calculation framework of the MDEM consists of two parts: particle motion equations and particle internal forces, which is briefly MDEM consists of two parts: particle motion equations and particle internal forces, which is briefly elaborated below. elaborated below.
Figure Figure1.1.Discretization Discretizationprocedure procedureof ofaaplane planesteel steelframe. frame.
2.1.1. Particle Motion Equations The motion of each particle follows the Newton’s second law at any time t. If damping term is supplemented to make structures static, the motion equations for an arbitrarily particle can be expressed as .. . MU(t) + C M U(t) = Fext (t) + Fint (t), (1)
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..
Iθ(t) + C I θ(t) = Mext (t) + Mint (t),
(2)
where M and I are the equivalent mass and moment of inertia lumped at an arbitrarily particle, .. .. respectively; U and θ denote translational and rotational acceleration vectors in the global coordinate . . system, respectively; U and θ indicate translational and rotational velocity vectors in the global coordinate system, respectively; Fint and Mint are internal force and moment vectors acting at an arbitrarily particle, respectively; Fext and Mext external force and moment vectors acting at an arbitrarily particle, respectively; and C M and C I denote translational and rotational damping matrix, respectively. Additionally, C M and C I equal αM and αI in the dynamic analyses respectively while both are virtual terms under static loading. α is mass proportional damping coefficient. If the central finite difference algorithm is employed to solve Equations (1) and (2), the translational velocity and acceleration of a particle can be approximated as .
U ( n + 1) − U ( n − 1) , 2∆t
(3)
U (n + 1) − 2U (n) + U (n − 1) , ∆t2
(4)
U (n) = ..
U (n) =
where U (n − 1), U (n) and U (n + 1) are the displacements of a particle at step n − 1, n and n + 1, respectively; and ∆t is a constant time increment. Substituting Equations (3) and (4) into Equation (1) yields U ( n + 1) =
2∆t2 2 + α∆t
F ext + Fint 4 −2 + α∆t + U (n) + U ( n − 1), M 2 + α∆t 2 + α∆t
(5)
Equation (5) is the formula of the translational displacements of a particle. 2.1.2. Particle Internal Forces From the discretization procedure of a three-dimension steel frame in Figure 1, it can be found that the establishment of MDEM model is different from that of the conventional DEM model. In the MDEM, the radius of the particles at both ends (R B ) of all the members is first given, and then the radii of the particles in the members (R A ) are determined via equally dividing the remaining lengths of these members. The internal forces arising at the contact points (e.g., Point C) between connected particles (e.g., Particles A and B) need be first calculated, and then they are exerted reversely to these particles (e.g., Particles A and B), thereby acquiring internal forces of an arbitrary particle in Figure 1. The internal force and moment increment at a contact point can be obtained by contact constitutive model, which are ( ∆f(n)= k∆u(n) , (6) ∆m(n) = kθ ∆θ0 (n) where ∆f and ∆m are the incremental internal force and moment of inertia arising at a contact point, respectively; k and kθ indicate translational and rotational contact stiffness coefficients, respectively; and ∆u and ∆θ0 denote pure translational and rotational displacements of a contact point in a time step ∆t respectively, which are in the local coordinate system. The key point of the DEM is the determination of contact stiffness coefficients (i.e., k and kθ ) in Equation (6). The coefficients are obtained mainly from experiences or experiments for the conventional DEM, while the MDEM gives the formulas of contact stiffness coefficients. Prof. Ye team [22] supplemented rotational springs into contact constitutive model (Figure 1), which make that the force state of each contact element (i.e., equivalent beam AB in Figure 1) is the same as that of the spatial beam element with three translational and three rotational degrees of freedoms in the FEM.
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According to the strain energy equivalent principle and simple beam theory, the formulas of contact stiffness coefficients for three-dimension member structures can be expressed as [24] k=
EA/l 12EIy /l 3 12EIz
/l 3
kθ =
GIx /l EIy /l EIz /l
,
(7)
where l is the distance between the two particle centers; A, E and G denote the area, elastic modulus and modulus of shearing of each member, respectively; and Ix , Iy and Iz are the cross sectional moments of inertia of a member about the x, y and z axes, respectively. 2.2. MDEM for Modelling Geometric Nonlinearity Structural geometric nonlinearity comprises rigid body motion, large rotation and large deformation. For the FEM, special treatments and complex modifications are necessary to deal with problems of this kind, such as the arc-length method and co-rotational coordinate approach [27]. However, in the MFEM, particles are assumed to be rigid, and thus rigid body motion of particles can be removed naturally in the process of solving the pure displacement of contact points using the rigid body kinematics. Furthermore, the movement of each particle is represented numerically by a time-stepping algorithm. In the algorithm, the assumption of small deformation is satisfied within each time step, which makes that large deformation and large rotation issues are naturally handled [28]. Thus, the unified procedure is employed in the MDEM for geometric linear and nonlinear problems. 2.3. MDEM for Modeling Facture Behavior Fracture behavior possesses strong discontinuity. In the FEM, fracture is modeled by defining birth-death elements or failure elements, which may cause the non-conservation of mass and convergence difficulty [26]. Nevertheless, in the MDEM, a structure is represented by a set of finite rigid particle rather than by continuum, and each particle is in a dynamic equilibrium. Given an appropriate fracture criterion only, Fracture behavior can be simulated [22]. The fracture is macroscopically represented by the separate of two adjacent particles in Figure 1. The Bending strain is take as the fracture criteria in this study, which is denoted as
|ε θt | ≥ ε θ p ,
(8)
where ε θt and ε θ p are the bending strain at time t and ultimate bending strain, respectively. The ultimate bending strain is set to be 0.0004 according to relative experiments and studies [29]. 3. Member Discrete Element Modeling for Semi-Rigid Connections 3.1. Virtual Zero-Length Spring Element In this study, a semi-rigid joint is simulated by a multi-degree-of-freedom spring element with two particles but without actual mass and length. The basic concept of the spring element is similar to that of References [13,18]. The moment-rotation relationship of the zero-length spring element is expressed as a diagonal tangent stiffness matrix in other numerical approaches taking the FEM as a representation. In addition, the stiffness matrix needs to be integrated with structural total stiffness matrix. However, in the MDEM, the zero-length spring element is not directly involved in the calculation, but is employed only to modify the stiffness coefficient of each contact element adjacent to the semi-rigid connection. In addition, then the modified stiffness coefficient of the contact element is directly applied in the next time step. The whole procedure is simple and effective. Furthermore, computational time of the MDEM after considering semi-rigid connections has no increase. This is not only the nature difference between the zero-length spring element of the MDEM and that of other methods, but also the advantage of the MDEM processing semi-rigid connection problem.
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3. Member Discrete Element Modeling for Semi-Rigid Connections 3.1. Virtual Zero-Length Spring Element
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In this study, a semi-rigid joint is simulated by a multi-degree-of-freedom spring element with two particles but without actual mass and length. The basic concept of the spring element is similar to that of References [13,18]. The moment-rotation relationship of the zero-length spring element is expressed as a diagonal tangent stiffness matrix in other numerical approaches taking the FEM as a representation. In addition, the stiffness matrix needs to be integrated with structural total stiffness matrix. However, in the MDEM, the zero-length spring element is not directly involved in the calculation, but is employed only to modify the stiffness coefficient of each contact element adjacent to the semi-rigid connection. In addition, then the modified stiffness coefficient of the contact element is directly applied in the next time step. The whole procedure is simple and effective. Furthermore, computational time of the MDEM after considering semi-rigid connections has no increase. This is not only the nature difference between the zero-length spring element of the MDEM and that of other methods, but also the advantage of the MDEM processing semi-rigid connection problem. The MDEM model of a simple frame is displayed in Figure 2. If the joint D (i.e., particle D) is a semi-rigid connection, the processing mode of the semi-rigid connection is given in Figure 3, that is, a zero-length spring element is supplemented at the particle. The spring element contains two x θx particles E and F with same coordinates, and its translational and rotational stiffness matrix can be expressed s as y sθ θy
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The MDEM model of a simple frame is displayed in Figure 2. If the joint D (i.e., particle D) is a semi-rigid connection, the processing mode of the semi-rigid connection is given in Figure 3, that is, a zero-length spring element is supplemented at the particle. The spring element contains two particles E and F with same coordinates, and its translational and rotational stiffness matrix can be expressed as
k =
R
R
R R z
k
R
=
R
Rθ x x Ry Rθ y ks = ksθ = , Rz Rvectors translational and rotational stiffness θz
Rθz
,
(9)
(9)
where Rn and Rθn are the of a zero-length spring element with respect to n axis (n = x, y, z). In addition, for a two-dimensional problem, the zero-length spring where R n and Rθ n are the translational and rotational stiffness vectors of a zero-length spring element can degradeelement to three degrees of= x,freedom which is two translational degrees of freedom and with respect to n axis (n y, z). In addition, for a two-dimensional problem, the zero-length spring element can degrade to three degrees of freedom which is two translational degrees of one rotational degree of freedom. freedom and one rotational degree of freedom.
Appl. Sci. 2017, 7, 714 7 of 24 Figure 2. MDEM (Member Discrete Element Method) of frame. a simple steel frame. Figure 2. MDEM (Member Discrete Element Method) model ofmodel a simple steel
Figure 3. Zero-length spring element at a semi-rigid connection. Figure 3. Zero-length spring element at a semi-rigid connection.
The comparison of Figures 1 and 3 shows that the force state of the zero-length spring element is identical with that of the1contact former is dummy the length spring of the The comparison of Figures and 3element, showsexcept that that the the force state of thewhile zero-length element is latter is related to connected particles. This characteristic makes that the MDEM is able to simulate a identical with that of the contact element, except that the former is dummy while the length of the semi-rigid connection by directly modifying the contact element stiffness. Only the bendingparticles. stiffness of aThis semi-rigid joint is frequently taken intothe account in steel latter is related to connected characteristic makes that MDEM isframes. able to simulate a The relationship between the moment andthe the contact relative rotation of astiffness. semi-rigid connection can be semi-rigid connection by directly modifying element expressed as follow: Only the bending stiffness of a semi-rigid joint is frequently taken into account in steel frames. The relationship between the moment and the can be ΔM relative = Rθ Δθr , rotation of a semi-rigid connection (10) expressed as follow: where Rθ is the linear or nonlinear stiffness∆M of a semi-rigid (i.e., zero-length spring element = Rθ ∆θconnection (10) r, stiffness), which is obtained by experiments or mathematic models; Δθr denotes the relative relation where Rθ is the linear or nonlinear between particles E and F. stiffness of a semi-rigid connection (i.e., zero-length spring element the semi-rigid connection of this in Figure 2 asmodels; an example, derivationthe detail of stiffness), whichTaking is obtained by experiments ortype mathematic ∆θthe relative relation r denotes the rotational stiffness coefficient of the contact element 1 is given. When the joint D is rigid, between particles E and F. the rotational stiffness of the members 1 and 2 are Kθ1 = E1I1 / L1 and Kθ 2 = E2I2 / L2 , respectively, L1 Taking the semi-rigid connection of this type in Figure 2 as an example, the derivation detail of the and L are the lengths of the members 1 and 2. When the joint D is semi-rigid, the semi-rigid rotational stiffness2 coefficient of the contact element 1 is given. When the joint D is rigid, the rotational connection stiffness is assumed to be linearly distributed to the contact elements 1 and 2 in accordance stiffness of the members 1 of and are Kθ 1 and = E2.1 Therefore, I1 /L1 and L1 and L2 are the 2 , respectively, θ 2 = E2 I2 /L with the stiffness the 2members theK contributions of the semi-rigid connection lengths of the 1 and 2.1 When theKjoint D is semi-rigid, the semi-rigid connection stiffness is to members the contact elements and 2 are θ2 ⋅ Rθ / (Kθ1 + Kθ2 ) and Kθ1 ⋅ Rθ / (Kθ1 + Kθ2 ) , respectively, denoted m1 and m2 . Assuming that two particles of the contact element 1 have a relative rotation Δ θ
, the deformation energy caused by the relative rotation is
1 1 Jcθ = kθ' 1 (Δθ )2 + m1 (Δθ )2 , 2 2 ' where kθ 1 is the modified rotational stiffness coefficient of a contact element.
(11)
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assumed to be linearly distributed to the contact elements 1 and 2 in accordance with the stiffness of the members 1 and 2. Therefore, the contributions of the semi-rigid connection to the contact elements 1 and 2 are Kθ2 · Rθ /(Kθ1 + Kθ2 ) and Kθ1 · Rθ /(Kθ1 + Kθ2 ), respectively, denoted m1 and m2 . Assuming that two particles of the contact element 1 have a relative rotation ∆θ, the deformation energy caused by the relative rotation is 1 1 Jcθ = k0θ1 (∆θ )2 + m1 (∆θ )2 , (11) 2 2 where k0θ1 is the modified rotational stiffness coefficient of a contact element. ∆θ is so small in a time step that the length of the contact element 1can still be considered equal to the distance of two particle centers. The strain energy produced by pure bending deformation of the equivalent beam composed of two particles is expressed as Jbθ =
Z L 1 0
2
E1 I1 κ 2 dx =
Z L 1 0
2
E1 I1 (
1 E1 I1 ∆θ 2 ) dx = (∆θ )2 , l1 2 l1
(12)
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When the joint is2017, semi-rigid, the rotational stiffness coefficient of the contact element 1 is obtained Appl.DSci. 7, 714 8 of 24 Appl. Sci. 2017, 7, 714 8 of 24 according to the equivalent principle of strain energy (i.e., Jcθ = Jbθ ), which is
EI kθ'' 1 = E11 I11 − m1 = kθ 1 − m1 , (13) E k = m1 = kθ 1 − m1 , E I θ' 1 = l11I1 −− m (13) = k − m , (13) k0θ1 = 1 1 k− k − m , (13) θ 1 m1 l= θ 1 1 1 1 l11 θ1 l1 where kθ 1 is the rotational stiffness coefficient of the contact element 1 when the joint D is rigid. where kθ 1 is the rotational stiffness coefficient of the contact element 1 when the joint D is rigid. rotational stiffness coefficient of the contact element 1the when the D same is rigid. where k θ1 is where the rotational stiffness coefficient of the contact element joint Djoint is the rigid. Thekθrotational stiffness coefficient of the contact element1 2when can be obtained in way. 1 is the The rotational stiffness coefficient of the contact element 2 can be obtained in the same way. The rotational stiffness coefficient of the contact element 2 can be obtained in the same way. In the MDEM, Equation (13) is the general formula for modifying rotational stiffness of the contact The rotational stiffness coefficient of the contact element 2 can be obtained in the same way. In the In the MDEM, Equation (13) is the general formula for modifying rotational stiffness of the contact In the (13) MDEM, Equation is the general for commonly modifying rotational ofelements the contact elements atisthe semi-rigid connections. For formula three most used semi-rigid connection types MDEM, Equation the general(13) formula for modifying rotational stiffness of thestiffness contact elements at the semi-rigid connections. For three most commonly used semi-rigid connection types for steel structures, specific formulas for modifying rotational stiffness are listed in Table 1. elements at the semi-rigid connections. For three most commonly used semi-rigid connection types at the semi-rigid connections. For three most for commonly semi-rigid connection for steel structures, specific formulas modifyingused rotational stiffness are listed intypes Tablefor 1. steel for steel structures, specific formulas for modifying rotational stiffness are listed in Table 1.
structures, specific formulas for modifying rotational stiffness are listed in Table 1.
Table 1. Specific formulas for modifying rotational stiffness for three semi-rigid connection types. Table 1. Specific formulas for modifying rotational stiffness for three semi-rigid connection types. Table 1. Specific formulas for modifying rotational stiffness for three semi-rigid connection types.
Semi-Rigid Table 1. Specific formulas for modifying rotational stiffness for three semi-rigid connection types. Configuration Formulas Semi-Rigid Connection Types Semi-Rigid Configuration Formulas Configuration Formulas Connection Types ConnectionTypes Types Semi-Rigid Connection Configuration Formulas Elastic support Elastic support Elastic Elastic supportsupport
kθ'' = kθ - 1/ 2 Rθ k' = k - 1/ 2 Rθ 1/ 2 R kk0θ θθ==k θkθθ−-1/2R θθ
Couple-bar joint Couple-bar joint Couple-bar joint Couple-bar joint
k0θ θ' ==kθkθ−-1/2R 1/ 2 R kk θθ = kθsame - 1/ 2bars) R (for θ two θ (for (fortwo twosame samebars) bars)
Beam-column joint Beam-column joint
k0θkθ'==RR θθ k =R (forsemi-rigid semi-rigid connections thethe beam) kθθ' = Rθθ onon (for connections beam)
kθ'' = kθ - 1/ 2 Rθ
(for two same bars) '
Beam-column joint Beam-column joint
(for semi-rigid connections on the beam) (for semi-rigid connections on the beam)
3.2.Connection Semi-Rigid Connection 3.2. Semi-Rigid Models Models 3.2. Semi-Rigid Connection Models 3.2. Semi-Rigid Connection Models Rsθthe Equation (10) is the general of formula of semi-rigid connection models. When in the formula Equation (10) is the general formula semi-rigid connection models. When Rsθ in formula Rsθ in the Equation (10) is the general formula of semi-rigid connection models. When formula R Equation (10) is the general formula of semi-rigid connection models. When in the formula is a variable, the semi-rigid connection model is nonlinear. Two different models are employed s θ is a variable, the semi-rigid connection model is nonlinear. Two different models are employed to to is a variable, the semi-rigid connection model is nonlinear. Two different models are employed to is a variable, the semi-rigid connection modelconnections is nonlinear. Two models to capture the nonlinear behaviors of semi-rigid this different study. The first isemployed the bilinear capture the nonlinear of semi-rigid connections in thisin The first one one isare the bilinear capture thebehaviors nonlinear behaviors of semi-rigid connections instudy. this study. The first one is the bilinear capture the nonlinear behaviors of semi-rigid connections in this study. The first one is the bilinear semi-rigid connection model presented by Keulen et al. [8]. The model is adopted in the classic semi-rigid connection model presented by Keulen et al. [8]. The is adopted in the classic semi-rigid connection model presented by Keulen et al. [8].model The model is adopted in the classic semi-rigid presented Keulen proposed. et al. [8]. The modelthe is adopted examples toconnection verify the model correctness of theby approach However, accuracy in of the the classic model examples toexamples verify the of the approach proposed. However, the accuracy of the model to correctness verify the correctness of the approach proposed. However, the accuracy of the model examples to verify the correctness approach thenonlinear accuracybehaviors of the model can only meet engineering practice.of Inthe order to moreproposed. accuratelyHowever, capture the and can only meet practice.practice. In order to more accurately capture behaviors canengineering only meet engineering In order to more accurately capturethe thenonlinear nonlinear behaviors and energy of semi-rigid connections, number of nonlinear models have been can onlydissipation meet engineering practice. In order toamore accurately capturesemi-rigid the nonlinear behaviors and energy dissipation of semi-rigid connections, a numberofofnonlinear nonlinear semi-rigid models have been and energy energy dissipation of semi-rigid connections, a number semi-rigid models have dissipation ofthe semi-rigid connections, are a number of nonlinear semi-rigid models developed. Of these, most representatives the Kishi–Chen three-parameter power have modelbeen [4], developed. Of these, the most representatives are thethe Kishi–Chen three-parameter power model [4], been developed. Of these, thethe most are Kishi–Chen three-parameter power developed. Of these, mostrepresentatives representatives are the three-parameter power the Richard–Abbott four-parameter model [5], theKishi–Chen Chen–Lui exponential model [9],model and [4], the the Richard–Abbott four-parameter model [5], the Chen–Lui exponential model [9], and the model [4], the Richard–Abbott four-parameter model [5],the the Chen–Lui Chen–Lui exponential model [9],and and Ramberg–Osgood model [30]. The results of Chan and Chui [14] asexponential well as Nguyen and[9], Kim [Error! the Richard–Abbott four-parameter model [5], model the Ramberg–Osgood model [30]. The results of Chan and Chui [14] as well as Nguyen and Kim [Error! the Ramberg–Osgood [30]. The ofthe Chan and Chui [14] as as well asasNguyen Kim [19] is Ramberg–Osgood model [30].results Thethat results of Chan and Chui [14] well Nguyenand and Kim [Error! Bookmark model not defined.] indicate effect of different models on the behavior of steel frames Bookmark not defined.] indicate that the effect of different models on the behavior of steel frames is Bookmark not defined.] the effect of different models model on the [5] behavior of steel frames is not significant. Therefore,indicate only thethat Richard–Abbott four-parameter is applied in this study, not significant. Therefore, only the Richard–Abbott four-parameter model [5] is applied in this study, not Therefore, onlybethe Richard–Abbott four-parameter model [5] is applied in this study, the significant. formula of the model can expressed as the formula of the model can be expressed as the formula of the model can be expressed as ( Rki − Rkp ) ( Rki − Rkp ) M = + Rkp θ r , 1/ n ( Rki− − R)kpθ) M = + Rkp θ r , ( R R (14) n 1/ ki kp r θr , M = 1 + ( ( R − R ) θ ) n 1/ n + Rkp (14) ki kp r n θ ( R − R ) (14) 1 + ( ) M ki kp r 0 ) n M0 1 + (
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indicate that the effect of different models on the behavior of steel frames is not significant. Therefore, only the Richard–Abbott four-parameter model [5] is applied in this study, the formula of the model can be expressed as ( Rki − Rkp ) M= (14) 1/n + Rkp |θr |, 1+(
( Rki − Rkp )|θr | n ) M0
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where M and θr are the moment and rotation of the semi-rigid connection, respectively; Rki , Rkp and M0 are theBehavior initial stiffness, theofstrain-hardening stiffness and the reference moment of the semi-rigid 3.3. Cyclic Modelling Semi-Rigid Connections connection, respectively, which are acquired by experiments [5]; n is the parameter defining the shape. The independent hardening model shown in Figure 4 is commonly used to model the cyclic behavior semi-rigid connections under cyclic loading because of its simple implementation 3.3. Cyclic of Behavior Modelling of Semi-Rigid Connections [13,Error! Bookmark not defined.,Error! Bookmark not defined.], which is also employed in this The independent hardening model shown in Figure 4 is commonly used to model the cyclic study. The virgin moment-rotation curve is shaped by the connection model in Equation (14). The behavior of semi-rigid connections under cyclic loading because of its simple implementation [13,18,21], instantaneous tangent stiffness is defined by taking derivative of Equation (14), which is expressed which is also employed in this study. The virgin moment-rotation curve is shaped by the connection as follow: model in Equation (14). The instantaneous tangent stiffness is defined by taking derivative of Equation (14), which is expressed as follow: 1/ n (15) Rkt = (d M dθr )t = (Rki − Rkp ) 1 + (Rki − Rkp ) ⋅ θr M 0 )n + Rkp , h i1/n Rkt = (dM/dθr )t = ( Rki − Rkp )/ 1 + ( Rki − Rkp ) · |θr |/M0 )n + Rkp , (15) The specific loading-unloading criteria of the model are detailed in References [Error! Bookmark not defined.,Error! Bookmark not defined.,Error! Bookmark not defined.]. The specific loading-unloading criteria of the model are detailed in [13,18,21]. M A
E Rkt
Rki Rki
D O
B
θr
C
Figure Independent hardening Figure 4. 4. Independent hardening model. model.
3.4. Computational Procedures for Static and Dynamic Dynamic Analysis Analysis of of Steel Steel Frames Frames with with Semi-Rigid Semi-Rigid Joints Joints The computational procedure of the MDEM is concise concise and and straight-forward. straight-forward. From the particle motion equations (i.e., Equations (1) and (2)), it can be found that each particle is in dynamic dynamic equilibrium, equilibrium, and and the difference difference between between static static analysis analysis and dynamic dynamic analysis analysis lies in the determination of damping. Therefore, Therefore, the the computational computational framework framework of of static static analysis analysis is consistent with that of dynamic analysis, and the unified MDEM flowchart for static and dynamic analysis of steel frames with is displayed in Figure 5. Additionally, all examples in this study were computed with semi-rigid semi-rigidjoints joints is displayed in Figure 5. Additionally, all examples in this study were by FORTRAN language. In addition,Inthe computational hardware environment is 4.0 GHz intel8-core computed by FORTRAN language. addition, the computational hardware environment is 4.0 GHz CPU, 16 G RAM hard disk. hard disk. intel8-core CPU, and 16 GST2000DM001-1ER164 RAM and ST2000DM001-1ER164
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1. Input initial structural parameters, material properties and so on 2. Assemble element stiffness matrix and lumped mass matrix 3. Calculate contact stiffness coefficients between two adjacent particles according to Eq.(7)
Yes
Consider semirigid connections ? No
Modify the stiffness coefficients of contact elements at the semi-rigid connections according Eq.(13)
1. Solve motion equations of particles (i.e. Eqs.(1) and (2) at time t=0 2. Store particle displacement, velocity and acceleration, move to next time step t + Δt Update particle coordinate, calculate the pure deformation (i.e. Δu and Δθ' ) of contact points by using the rigid body kinematics
Are semi-rigid connections nonlinear ?
Yes
No Calculate the internal forces of contact element according to Eq.(6) (i.e. contact constitutive model) No
Reach the fracture criteria (Eq.(8) ?
Yes
1. Calculate the instantaneous tangent stiffness of semi-rigid connections by Eq.(15) , and obtain the modified contact element stiffness at time t + Δt by substituting it into Eq.(13) 2. Determine the force state of each semi-rigid connection according to independent hardening model in section 2.3 The internal forces of the contact element at the fracture are set to be zero
No 1. Calculate particle internal forces by exerted reversely internal forces of the contact element to connected particles 2. Substitute particle internal forces into Eqs.(1) and (2), and store particle displacement, velocity and acceleration at the time step t + Δt
t >T End
Figure MDEMflowchart flowchartfor for static static and ofof steel frames with semi-rigid connections. Figure 5. 5. MDEM anddynamic dynamicanalysis analysis steel frames with semi-rigid connections.
4. Examples
4. Examples
The applicability and accuracy of the procedure proposed are verified by various structural The applicability and nonlinearity, accuracy ofsnap-through the procedure proposed areresponse verifiedand byfracture. various behaviors such as geometric buckling, dynamic Ofstructural these, behaviors such as geometric nonlinearity, snap-through buckling, dynamic response and the beam-column joint is assumed to be linear (e.g., in Examples 3.1–3.4), bilinear (e.g., in Examplesfracture. 3.5) Of orthese, the semi-rigid beam-column joint (e.g., is assumed to be (e.g., in 3.1–3.4), bilinear nonlinear connection in Examples 3.6)linear accordingly. TheExamples geometrically nonlinear (e.g., in Examples 3.5) orwith nonlinear semi-rigid (e.g., in Examples 3.6) accordingly. analysis of a column elastic support underconnection static loading is performed in Example 3.1 to testnonlinear the correctness of the procedure. static under analysis of a loading beam with elastic Thepreliminarily geometrically analysis of aproposed column with elasticThe support static is performed supports is carried out intest Example 3.2, and of then of this is in Example 3.1first to preliminarily the correctness thedynamic proposedresponse procedure. Thestructure static analysis to emphasize theis accuracy of the procedure dynamic loading. of supplemented a beam with elastic supports first carried outproposed in Example 3.2, andunder then dynamic response of 3.3isfurther employedto the proposed procedure intoofthe static and dynamic response of dynamic steel thisExample structure supplemented emphasize the accuracy the proposed procedure under frames with linear semi-rigid connections to validate the applicability of the proposed procedure in of loading. Example 3.3 further employed the proposed procedure into the static and dynamic response steel frames. In Example 3.4, the snap-through buckling behavior of steel frames is investigated in steel frames with linear semi-rigid connections to validate the applicability of the proposed procedure detail to reflect the advantages of the proposed procedure. Different from the above examples, in steel frames. In Example 3.4, the snap-through buckling behavior of steel frames is investigated Example 3.5 introduces the bilinear semi-rigid connection model proposed by Keulen et al. [8] into in detail to reflect the advantages of the proposed procedure. Different from the above examples, the MDEM for the static analysis of single-story and three-story frames with semi-rigid connections. Example 3.5 introduces the bilinear semi-rigid connection model proposed by Keulen et al. [8] into In order to capture the nonlinear behavior and hysteresis performance of semi-rigid connections themore MDEM for the static analysis3.6 of adopts single-story and three-story frames with semi-rigid connections. accurately, the Example the Richard-Abbort four-parameter model [5] given in In order to capture the nonlinear behavior and hysteresis performance of semi-rigid connections Section 2.2 for dynamic analysis of the double-span, six-story Vogel steel frame under cyclic loading.more accurately, thethe Example adopts the Richard-Abbort four-parameter model [5] given Section 2.2 In addition, effect of3.6 semi-rigid connections on the collapse behavior of the structure is in studied.
for dynamic analysis of the double-span, six-story Vogel steel frame under cyclic loading. In addition, the effect of semi-rigid connections on the collapse behavior of the structure is studied.
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4.1. Geometrically Nonlinear Analysis of a Column with Elastic Support
4.1.column Geometrically Nonlinear Analysis of a Column used with Elastic Support A with elastic support is frequently as a benchmark problem to verify the correctness of the approaches semi-rigid nonlinear analysis this structure A columnmodelling with elastic support connections. is frequently Geometrically used as a benchmark problem toofverify the correctness of the approaches modelling semi-rigid connections. Geometrically nonlinear analysis of has been performed by many researchers (Zhou and Chan [12]; Lien et al. [20]) using different methods. this structure has6,been researchers (Zhoubelow: and Chan [12]; Lien etAal.=[20]) using As shown in Figure the performed propertiesby of many the column are listed cross section 0.1 m × 0.1 m, different shown in Figure E6, = the210 properties of the column are listed below: length L = 3.2methods. m, andAs elastic modulus GPa. An axial concentrated load pcross wassection applied to A = 0.1 m × 0.1 m, length L = 3.2 m, and elastic modulus E = 210 GPa. An axial concentrated load p the column top (i.e., the free end), meanwhile a lateral load 0.01p was added in the consideration of was applied to the column top (i.e., the free end), meanwhile a lateral load 0.01p was added in the geometric imperfection. The material was assumed to be perfectly elastic. The support in the column consideration of geometric imperfection. The material was assumed to be perfectly elastic. The bottom was set as rigid and semi-rigid in the rotational direction, respectively. The rotation-resistant support in the column bottom was set as rigid and semi-rigid in the rotational direction, respectively. stiffness of the semi-rigid support 10EI/L. The krotation-resistant stiffness k ofisthe semi-rigid support is 10EI/L. In the MDEM, the column was discretized 11 particles particles(i.e., (i.e., elements). When the support In the MDEM, the column was discretized into into 11 1010 elements). When the support is semi-rigid, the stiffness of the bottom contact directionisisdetermined determined is semi-rigid, the stiffness of the bottom contactelement elementin inthe the rotational rotational direction to to be EI/l be − 5EI/L to Table 1, where l equals betweenthe the centers of two particles EI l −5according EI L according to Table 1, where l equalsthe thedistance distance between centers of two particles of theofcontact element. Figure 6 indicates resultsobtained obtained using modified MDEM the contact element. Figure 6 indicatesthat that the the results using thethe modified MDEM agreeagree well with the existing results (Zhou Lienetetal. al.[20]) [20]) two cases of fixed end and well with the existing results (Zhouand andChan Chan [12]; [12]; Lien inin thethe two cases of fixed end and semi-rigid supports. That also illustratesthat that the the modified can accurately andand effectively deal deal semi-rigid supports. That also illustrates modifiedMDEM MDEM can accurately effectively the geometrically nonlinearity column with with elastic with with the geometrically nonlinearity ofofa acolumn elasticsupport. support. 10
P,v 0.01P,u
8
l=3.2m A=0.1m×0.1m E=210GPa
MDEM model
k
2
PL /EI
6
4
Rigid end v- Zhou & Chan u-Zhou & Chan R-Zhou & Chan v-MDEM u-MDEM R-MDEM
2
0 0
1
2
3
4
Semi-rigid end Lien et al u-Zhou & Chan R-Zhou & Chan u-MDEM R-MDEM v-MDEM 5
Displacement at the free end u/m,v/m,R/rad
6
Figure 6. Load-displacement curves curves at of of thethe column. Figure 6. Load-displacement atthe thefree freeend end column.
4.2. Static and Dynamic Response of a Beam with Elastic Ends
4.2. Static and Dynamic Response of a Beam with Elastic Ends The schematic configuration of a beam with elastic ends is displayed in Figure 7. The properties
The configuration a beam with elastic is displayed 7. and The elastic properties of theschematic beam are listed as follow: of cross section A = 25.4 mmends × 3.175 mm, lengthin L =Figure 508 mm, of themodulus beam are as follow: cross increasing section A =concentrated 25.4 mm ×load 3.175 length = mid-span 508 mm, and elastic E =listed 207 GPa. A gradually P mm, is applied at L the point. modulus E = 207 A[31] gradually increasing load Pthe is static applied the mid-span Mondkar and GPa. Powell as well as Yang and concentrated Saigal [32] analyzed andat dynamic responsepoint. of theand beam with fixed Theas static and dynamic of the beam with semi-rigid ends response was Mondkar Powell [31]ends. as well Yang and Saigalresponse [32] analyzed the static and dynamic investigated by Chui and Chan [13] and Lien et al. [20] using the conventional FEM and the VFIFE, of the beam with fixed ends. The static and dynamic response of the beam with semi-rigid ends was respectively. The rotation-resistant stiffness semi-rigid is EI/L. investigated by Chui and Chan [13] and Lienk of et the al. [20] usingends the conventional FEM and the VFIFE, In this study, the beam was represented by 51 particles (i.e., 50 contact elements). The ends were respectively. The rotation-resistant stiffness k of the semi-rigid ends is EI/L. modeled as rigid and semi-rigid in the rotational direction, respectively. The semi-rigid behavior was In this study, the beam was represented by 51 particles (i.e., 50 contact elements). The ends were simulated by the contact elements at the ends with the stiffness in the rotational direction of 49EI/L modeled as rigid and semi-rigid in the rotational direction, respectively. The semi-rigid behavior which was obtained according to Table 1. Figure 7 shows that the load-displacement curves of the was simulated by theare contact elements at the ends with the(Mondkar stiffnessand in the rotational direction modified MDEM consistent with the published results Powell [29]; Chui and of 49EI/L which was obtained according to Table 1. Figure 7 shows that the load-displacement curves of the modified MDEM are consistent with the published results (Mondkar and Powell [29]; Chui
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and Chan [13]; stiffness, Appl. Sci. 2017,Lien 7, 714 et al. [21]); the stiffness of the ends has a significant effect on structural 12 of 24 and the stiffness of the beam with semi-rigid ends is obviously reduced. In addition, the stiffness of Chan [13]; Lien et al. [21]); the stiffness of the ends has a significant effect on structural stiffness, and Chan [13]; Lien et al. [21]); the stiffness of the ends has a significant effect on structural stiffness, and the beam increases with an with increase of the ends load,isthe reason reduced. is that the reaction at theofsupports the stiffness of the beam semi-rigid obviously In tensile addition, the stiffness the the the stiffness the beam with semi-rigid endsthe is obviously reduced. In addition, theatstiffness of the beam increases with an increase of the load, reason is that the tensile reaction the supports induces axialoftension arising in the beam. beam increases with an increase thebeam. load, the reason is that the tensile reaction at the supports induces the axial tension arising inofthe The dynamic response analysis of the same beam was further performed. A sudden load was induces the axial tension arising in the beam. The dynamic response analysis of the same beam was further performed. A (640 sudden wasa very applied at the mid-span point, that is, the load p was linearly increased to pmax lb) load within Theatdynamic response analysis theload same beam was further performed. A sudden load was applied the mid-span point, that is,ofthe p was linearly increased to pmax (640 lb) within a very shortapplied time (10 µs),mid-span and then the p maintained until theincreased calculation ended (t = 5 s), as shown maxis,isthe at the point, that load p was linearly to p max (640 lb) within a very short time (10 μs), and then the pmax is maintained until the calculation ended ( t = 5 s ), as shown in in Figure During wasuntil assumed to be perfectly the viscous short 8. time (10 μs),the andanalysis, then the the pmax material is maintained the calculation ended ( elastic, as shown in t = 5 s ), and Figure 8. During the analysis, the material was assumed to be perfectly elastic, and the viscous damping was ignored. It can be found from Figure 8 that the analysis results of the modified MDEM is Figure 8. was During the analysis, the material was assumed be perfectly and the MDEM viscous damping ignored. It can be found from Figure 8 that theto analysis resultselastic, of the modified identical with published results, and the minor error is due to the inevitable numerical method itself. damping was ignored. It canresults, be found that is thedue analysis of the modifiedmethod MDEM is identical with published andfrom the Figure minor 8error to theresults inevitable numerical Moreover, the stiffness of the ends affects significantly on the period and amplitude of displacement is identical with published results, and the minor error is due to the inevitable numerical method itself. Moreover, the stiffness of the ends affects significantly on the period and amplitude of itself. Moreover, stiffness of the ends affects significantly on the with period and whereas amplitude of the time displacement history curves. The structural stiffness decreases with decreases semi-rigid supports, both timethe history curves. The structural stiffness semi-rigid supports, displacement time history curves. The structural stiffness decreases with semi-rigid supports, period and amplitude increase accordingly. whereas both the period and amplitude increase accordingly. whereas both the period and amplitude increase accordingly. 800 800
P, w P, w
k k
Load P/lb Load P/lb
600 600
k k L/2 L/2
L/2 L/2
3.175mm 3.175mm 400 400
Rigid end Rigid end Mondkar & Powell Mondkar & Powell Chui & Chan Chui & Chan Lien et al Lien et al MDEM MDEMend Semi-rigid Semi-rigid Chui &end Chan Chui et &alChan Lien Lien et al MDEM MDEM
25.4mm 25.4mm k=EI/L k=EI/L L=508mm L=508mm 2 E=207kN/mm E=207kN/mm2
200 200
0 00.0 0.0
0.1 0.1
0.2
0.3
0.4
0.5
0.6
0.2 0.3 0.4 0.5 w/inch0.6 Displacement at the mid-span point Displacement at the mid-span point w/inch
0.7 0.7
Figure 7. Displacement at the mid-span point of the beam under static loading.
Figure 7. Displacement pointofofthe thebeam beam under static loading. Figure 7. Displacementatatthe themid-span mid-span point under static loading.
Displacement mid-span point / inch Displacement at at thethe mid-span point / inch
1.0 1.0
Semi-rigid end Semi-rigid end Chan Chui and Chui and Lien et al Chan Lien et al MDEM MDEM
0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2
Rigid end Rigid and endPowell Mondkar
0.0 0.0
-0.2 -0.2 0 0
P P 640lb 640lb
Mondkar Powell Yang andand saigal Yangand andChan saigal Chui Chuiet and Lien al Chan Lien et al MDEM MDEM 1000 2000 1000 2000Time t
10 10 3000
/μs3000 Time t /μs
5000 5000 4000 4000
t/μs t/μs 5000 5000
Figure 8. Displacement time history curves at the mid-span point of the beam under sudden loading. Figure 8. Displacement time history curves at the mid-span point of the beam under sudden loading.
Figure 8. Displacement time history curves at the mid-span point of the beam under sudden loading.
The MDEM discretized the single-bay two-story steel frame into 28 particles (i.e., 28 contact elements). Of these, each column was represented by 4 contact elements while every beam by 6 contact elements. The bending-resistant stiffness of the contact elements at the beam ends was modified to be 3.48 × 107 kN-m/rad according to Table 1. The horizontal displacement of steel frames with linear semi-rigid joints is shown in Figure 9. The figure shows that the load-displacement Appl. Sci. 2017, 7, 714 12 of 23 relationship of the modified MDEM is in a good agree with the published results. Additionally, the effect of semi-rigid connections on the structural behavior is great, and structural loading capacity be and overestimated if semi-rigid connections are ignored. 4.3.may Static Dynamic Response Analysis of Steel Frames with Linear Semi-Rigid Connections Another single-bay two-story steel frame with linear semi-rigid joints was adopted to analyze 9 displays andloading loading of astructure single-bay two-story steel 10. frame with theFigure dynamic response.the Theconfiguration configuration and of this are shown in Figure All the W14 × 48 steel beams and W12 × 96 steel columns. The elastic modulus is 200 GPa. A vertical beams and columns are W8 × 48 and the Young’s modulus is 205 GPa. The material of all the frame concentrated load P was applied to fourelastic joints,and meanwhile a horizontal loadignored. 0.01p was members was assumed to be perfectly the viscous damping was Theadded initial to consider theimperfection geometric imperfection orbewind Several studies (Lui and Chen [9], Zhouset and geometric ψ was taken to 1/438load. rad. The lumped masses of 5.1 T and 10.2 T were Chan [12]; Lien et al. [20]) carried out stability analysis of the structure in the two cases of rigid and at the top of the columns and in the middle of the beams, respectively. The stiffness of semi-rigid semi-rigid connections using different methods. the above analysis, the bending-resistant connections was 23,000 kN-m/rad. In simulation this study, each beam In was simulated by 9 particles (i.e., 8 contact elements), while every column wasjoints discretized into 4toparticles elements). As the stiffness stiffness of semi-rigid beam-column was taken be 3.48 (3 × contact 107 kN-m/rad. of The eachMDEM frame discretized member is same, the bending-resistant stiffness the28contact elements the the single-bay two-story steel frame of into particles (i.e., 28atcontact semi-rigid joints can be determined to be 11,500 kN-m/rad by Equation (13) not Table 1. The vertical elements). Of these, each column was represented by 4 contact elements while every beam by 6 contact static loads 50 KN and 100 KN were first applied onelements the beam-column joints andwas in the middle to of be elements. Theof bending-resistant stiffness of the contact at the beam ends modified 7 beams to consider the second-order effects, and then the horizontal forces were added suddenly at 3.48 × 10 kN-m/rad according to Table 1. The horizontal displacement of steel frames with linear each floor during 0.5 s until the calculation ended. The horizontal displacement on the top is shown semi-rigid joints is shown in Figure 9. The figure shows that the load-displacement relationship of the in Figure 11. Theiscomparison indicates the displacement responses of the MDEM and existing results modified MDEM in a good agree with the published results. Additionally, the effect of semi-rigid (Chan and Chui [13], Nguyen and Kim [18]) are consistent, and the influence of semi-rigid connections on the structural behavior is great, and structural loading capacity may be overestimated connections on the periods and the amplitudes of displacement responses of steel frames is more if semi-rigid connections are ignored. significant as well as the amplitude increase is up to 60%. 800 700
Rigid joint Semi-rigid joint P
400 300
Lui and Chen Zhou and Chan Lien et al MDEM
Rigid
200
0.001P
P
Beams: W14×48 Columns:W12×96 P P
0.001P EI=constant
100
Zhou and Chan Lien et al MDEM-semirigid
Semi-rigid 0
3.66m
500
k=3.47e7kN-mm/rad
3.66m
Vertical load P/kips
600
6.1m
-100 0
2
4
6
8
10
12
14
Horizontal displacement /inch Figure9.9.Horizontal Horizontaldisplacement displacement of of steel steel frames Figure frameswith withlinear linearsemi-rigid semi-rigidjoints. joints.
Another single-bay two-story steel frame with linear semi-rigid joints was adopted to analyze the dynamic response. The configuration and loading of this structure are shown in Figure 10. All the beams and columns are W8 × 48 and the Young’s modulus is 205 GPa. The material of all the frame members was assumed to be perfectly elastic and the viscous damping was ignored. The initial geometric imperfection ψ was taken to be 1/438 rad. The lumped masses of 5.1 T and 10.2 T were set at the top of the columns and in the middle of the beams, respectively. The stiffness of semi-rigid connections was 23,000 kN-m/rad. In this study, each beam was simulated by 9 particles (i.e., 8 contact elements), while every column was discretized into 4 particles (3 contact elements). As the stiffness of each frame member is same, the bending-resistant stiffness of the contact elements at the semi-rigid joints can be determined to be 11,500 kN-m/rad by Equation (13) not Table 1. The vertical static loads of 50 KN and 100 KN were first applied on the beam-column joints and in the middle of beams to consider the second-order effects, and then the horizontal forces were added suddenly at each floor during 0.5 s until the calculation ended. The horizontal displacement on the top is shown in Figure 11. The comparison indicates the displacement responses of the MDEM and existing results (Chan and
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Chui [13], Nguyen and Kim [18]) are consistent, and the influence of semi-rigid connections on the periods and the amplitudes of displacement responses of steel frames is more significant as well as the Appl. Sci.increase 2017, 7, 714is up to 60%. 14 of 24 amplitude 50kN
100kN
50kN
0.5F(t) 50kN
100kN
50kN
3.0m
All the frame member:W8×48 E=205GPa Lumped mass： =5.1T =10.2T
F(t) 3.0m
Ψ=1/438
Ψ=1/438
F(t)
100KN
2×4.0=8.0m
t/s
0.5
Figure configurationand andloading loading of the in dynamic analysis. Figure 10. 10. TheThe configuration thestructure structureadopted adopted in dynamic analysis. 0.15
Displacement /m
0.10 0.05 0.00 -0.05 -0.10
Rigid
-0.15
Semi-rigid
Nguyen and Kim Chan and Chui MDEM
-0.20 0.0
0.5
1.0
Nguyen and Kim Chan and Chui MDEM 1.5
Time /s
2.0
2.5
3.0
Figure 11. Displacement time history steelframe framewith with linear semi-rigid connections. Figure 11. Displacement time historycurves curves of aa steel linear semi-rigid connections.
4.4. Snap-Through Buckling Analysisofofthe theWilliams Williams Toggle Toggle Frame Semi-Rigid Connections 4.4. Snap-Through Buckling Analysis Framewith withLinear Linear Semi-Rigid Connections and Supports and Supports The Williams Toggle frame is frequently taken as a typical example for snap-through buckling
The Williams Toggle frame frequentlytaken taken as forfor snap-through buckling The Williams Toggle frame is is frequently as aa typical typicalexample example snap-through buckling analysis. The structure has the obviously nonlinear bifurcation instability, that is, each bifurcation analysis. The structure has the obviously nonlinear bifurcation instability, that is, each bifurcation point point may have one or more equilibrium paths and each path represents the extreme point instability may have one or more [Error! equilibrium pathsnot and each path Many represents the extreme instability or snap or snap instability Bookmark defined.,33]. researchers (Zhou point and Chan [12], Lien instability [12,33]. Many researchers (Zhou and Chan [12], Lien et al. [20]) performed static stability et al. [20]) performed static stability analysis of the steel frame with linear semi-rigid connections and analysis of theand steel with linear stiffness semi-rigid connections supports, and thek bending-resistant supports, theframe bending-resistant of each semi-rigidand connection or support is 10EI/L. The stiffness of eachofsemi-rigid connection or support k is 10EI/L. The properties of the Williams Toggle properties the Williams Toggle frame is as follow: cross section A = 19.13 mm × 6.17 mm; the modulus E = section 71 GPa. A = 19.13 mm × 6.17 mm; the Young’s modulus E = 71 GPa. frameYoung’s is as follow: cross Negative or zero structural stiffnessmay may occur occur at buckling, resulting that that special Negative or zero structural stiffness atthe theinstant instantofof buckling, resulting special treatments for conventional FEM is necessary to investigate snap-through buckling behavior. In theIn the treatments for conventional FEM is necessary to investigate snap-through buckling behavior. FEM, the most commonly used approach is the arc-length method [26]. However, in the MDEM, no FEM, the most commonly used approach is the arc-length method [26]. However, in the MDEM, no negative or zero stiffness issues occur because of no assembly of stiffness matrix, and only
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negative or zero stiffness issues occur because of no assembly of stiffness matrix, and only displacement Sci. 2017, 7, 714 15 of 24 control Appl. method need be applied to trace the complete equilibrium path. In this study, each member was discretized intocontrol 13 particles contacttoelements), and the applied displacement increment displacement method (i.e., need 12 be applied trace the complete equilibrium path. In this study, was ∆ueach = 0.001 mm. The bending-resistant stiffness of the contact elements at the semi-rigid member was discretized into 13 particles (i.e., 12 contact elements), and the applied joints displacement was stiffness of theofcontact elementsand the was determined toincrement be 7EI/L by ΔTable buckling analyses the MDEM u = 0 .0 01. 1 mSnap-through m . The bending-resistant the semi-rigid was determined to be 7EI/L by result Table 1.[34]; Snap-through buckling of the existingatresults (Yangjoints and Chiou [31]; Experimental Zhou and Chan analyses [12]; Lien et al. [20]) MDEM and the existing results (Yang and Chiou [31]; Experimental resultMDEM [34]; Zhou and Chan [12]; carried are displayed in Figure 12. The comparison shows that the modified can accurately et al. [20]) are displayed in Figure 12. The comparison shows that the modified MDEM can out the Lien snap-through buckling analysis of the Williams Toggle frame with semi-rigid connections and accurately carried out the snap-through buckling analysis of the Williams Toggle frame with supports. Furthermore, semi-rigid connections and support have a significant effect on the critical load semi-rigid connections and supports. Furthermore, semi-rigid connections and support have a and post-buckling behavior. significant effect on the critical load and post-buckling behavior. 70
P 657.15mm
60
Yang & Chiou Eperimental Zhou & Chan Lien et al DEM
50
Load P
40
9.8mm
6.17mm 19.13mm
Rigid joint
30
20
Zhou & Chan E=71kN/mm2, k=10EI/L Lien et al L:member length MDEM
10
0 0.0
0.2
0.4
Zhou & Chan Lien et al MDEM 0.6
0.8
Displacement /in Figure 12. Displacement response of the Williams Toggle frame with linear semi-rigid connections
Figure 12. Displacement response of the Williams Toggle frame with linear semi-rigid connections and supports. and supports. 4.5. Static Analysis of a Steel Portal Frame with Bilinear Semi-Rigid Connections
4.5. Static Analysis of a Steel Portal Frame with Bilinear Semi-Rigid Connections The bilinear semi-rigid connection model presented by Keulen et al. [8] is adopted to simulate the nonlinear behavior of semi-rigid connections. Figure 13 shows the detail configuration of a steel The bilinear semi-rigid connection model presented by Keulen et al. [8] is adopted to simulate portal frame. The beam-column joints are bolted by flush endplates, which are obviously nonlinear the nonlinear behavior of semi-rigid connections. Figure 13 shows the detail configuration of a steel semi-rigid connections. Keulen et al. [8] carried out the second-order elastic-plastic analysis using the portal frame. beam-column joints are bolted by flush endplates, which are et obviously nonlinear ANSYSThe software (ANSYS, Inc., Pittsburgh, PA, USA). Del Savio et al. [35] and Lien al. [20] were semi-rigid connections. Keulen et al.using [8] carried the second-order further performed static analysis differentout numerical methods. elastic-plastic analysis using the The steel portal frame made of S335 IPE360 HEA260 [8].Lien The span ANSYS software (ANSYS, Inc.,was Pittsburgh, PA,steel, USA). Deland Savio et al.sections [35] and et al.and [20] were height were 7.2 m and 3.6 m, respectively. A concentrated load F was applied at the joints and in the further performed static analysis using different numerical methods. beam, as shown in Figure 13. The horizontal loads was set to be α F to consider the initial The steel portal frame was made of S335 steel, IPE360 and HEA260 sections [8]. The span geometrical imperfection and wind load, where α was taken to be 0.1, 0.15, 0.2, 0.3 and 0.5 and height were 7.2 m and 3.6 m, respectively. A concentrated load F was applied at the joints successively. The semi-rigid supports was assumed to be perfectly elastic-plastic, its initial stiffness and in is the beam, as shown inmoment Figurecapacity 13. The loads was set to be αF to consider the 23,000 kN-m/rad, and the washorizontal 200 kN-m. The moment-rotation relation of beaminitial geometrical and wind load, where α was14. taken to bewere 0.1,simulated 0.15, 0.2,using 0.3 and 0.5 column joints,imperfection proposed by Keulen et al. [8], is shown in Figure The joints the full characteristic nonlinear model and a bilinear model. The connection model applied in thisstiffness successively. The semi-rigid supports was assumed to be perfectly elastic-plastic, its initial study is the bilinear model in Figure 14. is 23,000 kN-m/rad, and the moment capacity was 200 kN-m. The moment-rotation relation of beam-column joints, proposed by Keulen et al. [8], is shown in Figure 14. The joints were simulated using the full characteristic nonlinear model and a bilinear model. The connection model applied in this study is the bilinear model in Figure 14.
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F W W F W W Beams: IPE360 k1 k1 Columns:HEA260 Beams: IPE360 k1 k1 Columns:HEA260 ：eaves joint k 1 W=F/6 kk21：base ：eavessupport joint W=F/6 E=210GPa k2：base support 2 k2 k2 FE=210GPa y=355N/m 2 k2 k2 Fy=355N/m 7.2m 7.2m
αF αF
3.6m 3.6m
F F
Figure 13. A steel portal frame with semi-rigid connections and supports. Figure 13. A steel portal frame with semi-rigid connections and supports. Figure 13. A steel portal frame with semi-rigid connections and supports. 300 300 250 250
Semi-rigid support Beam-column joint：Half initial secand stiffness Semi-rigid support Beam-column Beam-column joint joint： ：Full Halfcharacteristic initial secand stiffness Beam-column joint：Full characteristic
Moment /kN-m Moment /kN-m
200
Support Support
200 150 150 100 100 50 50
2/3Mj,Rd 2/3Mj,Rd
Moment capacity Mj,Rd=157.2 Moment capacity Mj,Rd=157.2
initial secant stiffness (30455e3N-m/rad) initial secant stiffness (30455e3N-m/rad) Half initial secant stiffness
(30455e3/2=15227.5e3N-m/rad) Half initial secant stiffness
0 (30455e3/2=15227.5e3N-m/rad) 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0
0.016 0.018 0.020
0.006 0.008 0.010/rad 0.012 0.014 Rotation
0.016 0.018 0.020
0.000 0.002 0.004
Rotation /rad
Figure 14. Load-moment curves. Figure 14. 14. Load-moment Load-moment curves. Figure curves.
In this study, there were 36 particles and 36 contact elements for this structure, and the radius of each wasthere 0.2 m. The36bending-resistant stiffnesselements of the contact at and the semi-rigid In particle this study, were particles and 36 contact for thiselements structure, the radius this study, there 36 particles andthat 36 contact elements for this and the radius joints was obtained by Equation (13), while of the contact atstructure, the supports could be of In each particle was 0.2were m. The bending-resistant stiffness of theelements contact elements at the semi-rigid ofdetermined each particle was 0.2 m. The bending-resistant stiffness of the contact elements at the semi-rigid accordingby toEquation Table 1. The comparison Figure 15 shows that thesupports modifiedcould MDEM joints was obtained (13), while thatgiven of theincontact elements at the be joints was obtained by to Equation (13),comparison while that of theincontact elements at supports the supports could be can accurately simulate the nonlinear behavior ofgiven semi-rigid connections of a MDEM singledetermined according Table 1. The Figure 15 showsand that the modified determined to Table 1. The the comparison in Figure 15 shows that the modified MDEM story steel according portal frame. Moreover, connection behavior obtained using simplified half initial can accurately simulate the nonlinear behavior ofgiven semi-rigid connections and supports of a singlecan accurately simulate the nonlinear behavior of semi-rigid connections and supports of a single-story secant stiffness approach a good the agreement withbehavior the full characteristic analysis. Taking = 0.2 story steel portal frame. has Moreover, connection obtained using simplified halfαinitial steel portal frame. Moreover, connection behavior usingin simplified as an example, the load-moment curves of joints and supports shown Figure 16.half It caninitial be α found secant stiffness approach has the a good agreement with theobtained full are characteristic analysis. Taking =secant 0.2 from figure the that thea semi-rigid beam-column joint oncharacteristic the right top first reached the plastic state, stiffness approach has good agreement full analysis. Taking α be = 0.2 as an as anthe example, load-moment curves of with joints the and supports are shown in Figure 16. It can found and plastic hinge froma the figure thatformed. the semi-rigid beam-column joint on the right top first reached example, the load-moment curves of joints and supports are shown in Figure 16.theItplastic can bestate, found andthe a plastic formed. from figurehinge that the semi-rigid beam-column joint on the right top first reached the plastic state, and a plastic hinge formed.
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Keulen et et al: al: Half Half initial initial secant secant stiffness stiffness Keulen Keulen et et al: al: Full Full characteristic characteristic Keulen Del Savio Savio Del MDEM MDEM
1200 1200
α=0.1 α=0.1
Load Load // kN kN
1000 1000
α=0.15 α=0.15
800 800
α=0.2 α=0.2 600 600
α=0.3 α=0.3
400 400
α=0.5 α=0.5
200 200
00
10 10
20 20
30 30
40 40
50 50
60 60
70 70
80 80
Horizontal displacement displacement on on the the top top //mm mm Horizontal
Figure 15. 15. Displacement response response on the the top of of the steel steel frame. Figure Figure 15. Displacement Displacement response on on the top top of the the steel frame. frame. 800 800 700 700 600 600
the joint joint on on the the left left top top the the joint joint on on the the right right top top the Support Support
Load Load /kN /kN
500 500 400 400 300 300 200 200
Plastic hinge hinge Plastic
100 100 00 -40 -40
-20 -20
00
20 20
40 40
60 60
80 80
100 100
Moment /kN-m /kN-m Moment
120 120
140 140
160 160
180 180
Figure 16. Load-moment curves of joints and supports. Figure 16. 16. Load-moment Load-moment curves curves of of joints joints and and supports. supports. Figure
analysisof three-storysteel steelportal portalframe framewas wasfurther furtherperformed, performed,as asshown shownin inFigure Figure17. 17. The static static analysis analysis of aa three-story three-story steel portal frame was further performed, as shown in Figure 17. The nonlinear behavior of the beam-column joint was simulated using the bilinear model simplified by The nonlinear behavior of the beam-column joint was simulated using the bilinear model simplified The nonlinear behavior of the beam-column joint was simulated using the bilinear model simplified thethe halfhalf initial secant stiffness approach. The procedure of determining the bending-resistant stiffness by the half initial secant stiffness approach. The procedure procedure of determining determining the bending-resistant bending-resistant by initial secant stiffness approach. The of the of contact elements at the semi-rigid connection was the same as the single-story frame. The horizontal stiffness of contact elements at the semi-rigid connection was the same as the single-story frame. stiffness of contact elements at the semi-rigid connection was the same as the single-story frame. load was set to 0.15F. Figure 18 shows the moment-rotation relation of the beam-column joint. The horizontal horizontal load load was was set set to to 0.15F 0.15F .. Figure Figure 18 18 shows shows the the moment-rotation moment-rotation relation relation of of the the beambeamThe The analysis of the MDEM the published results (Keulen et results al. [8] and Del Savio et[8] al. [33]) column joint.result The analysis analysis resultand of the MDEM and and the published published results (Keulen et al. al. [8] and column joint. The result of MDEM the (Keulen et and are displayed in Figure 19. The figure indicates that the modified MDEM possesses considerable Del Savio et al. [33]) are displayed in Figure 19. The figure indicates that the modified MDEM Del Savio et al. [33]) are displayed in Figure 19. The figure indicates that the modified MDEM accuracy for the nonlinear behavior of the semi-rigid connection of multi-story steel portal possesses considerable accuracy forsimulation the nonlinear nonlinear behavior simulation of the the semi-rigid connection connection possesses considerable accuracy for the behavior simulation of semi-rigid frames. Additionally, from the load-moment curves right joints curves and supports (Figure 20),and it is of multi-story steel portal portal frames. Additionally, fromon thethe load-moment curves on the the right right joints joints and of multi-story steel frames. Additionally, from the load-moment on found that for three-story steel portal frame with semi-rigid joints and supports, the beam-column supports (Figure 20), it is found that for three-story steel portal frame with semi-rigid joints and supports (Figure 20), it is found that for three-story steel portal frame with semi-rigid joints and joint on the side of first floor plastic state and a plastic the supports, theright beam-column joint onreached the right rightthe side of first first floor reached the hinge plasticformed. state and andWhen plastic supports, the beam-column joint on the side of floor reached the plastic state aa plastic load continues to increase to 336 kN and 403 kN accordingly, the plastic hinge in turn occurred at the hinge formed. When the load continues to increase to 336 kN and 403 kN accordingly, the plastic hinge formed. When the load continues to increase to 336 kN and 403 kN accordingly, the plastic beam-column joints on the right side of second floor and third floor. hinge in in turn turn occurred occurred at at the the beam-column beam-column joints joints on on the the right right side side of of second second floor floor and and third third floor. floor. hinge
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FF
W W
W W
0.15F 0.15F k k11
FF
3.6m 3.6m
kk1 1
W W
W W
0.15F 0.15F k k11
3.6m 3.6m
kk1 1
W W
Beams: Beams:IPE360 IPE360 Columns:HEA260 Columns:HEA260 W=F/6 W=F/6 α=0.15 α=0.15
kk2 2
kk1 1
joint kk1：eaves 1：eaves joint support kk2：base 2：base support E=210GPa E=210GPa2 kk2 FFy=355N/m 2 =355N/m2
3.6m 3.6m
0.15F 0.15F k k11
W W
y
7.2m 7.2m
Figure 17. A single-span three-story steel portal frame with semi-rigid connections and supports.
Figure 17.single-span A single-span three-storysteel steelportal portal frame connections andand supports. Figure 17. A three-story framewith withsemi-rigid semi-rigid connections supports. 350 350 300 300
Semi-rigid Semi-rigidsupport support Beam-column Beam-columnjoint joint： ：Half Half initial initialsecant secantstiffness stiffness Beam-column joint ： Full characteristic Beam-column joint：Full characteristic Moment =366.5 Momentcapacity capacityM Mj,Rd =366.5 j,Rd
250 250
Moment/kN-m /kN-m Moment
200 2/3Mj,Rd 200 2/3Mj,Rd 150 150 100 100 50 50
Half Halfinitial initialsecant secantstiffness stiffness (90700/2=45350kN-m/rad) (90700/2=45350kN-m/rad) Initial Initialsecant secantstiffness stiffness (90700kN-m/rad) (90700kN-m/rad)
0 0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020
Rotation Rotation/rad /rad
Figure 18. Load-moment curves. Figure curves. Figure18. 18.Load-moment Load-moment curves.
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Third floor floor Third
400 400 300 300 200 200 100 100 00
Second floor floor Second
Load LoadP/kN P/kN
400 400 300 300 200 200 100 100 00 400 400
First floor floor First
300 300
Keulen et et al: al: Full Full characteristic characteristic Keulen Keulen et al: Half initial secant secant stiffness stiffness Keulen et al: Half initial Del Savio Savio Del MDEM MDEM
200 200 100 100 00
00
50 50
100 100
150 150
200 200
Displacement /mm /mm Displacement
250 250
300 300
350 350
Figure 19. Load-Displacement curves curves at at each floor ofof the steel frame. Figure 19.19. Load-Displacement ateach eachfloor floor the steel frame. Figure Load-Displacement curves of the steel frame. 500 500
Right support support Right Right first first floor floor Right Right second second floor floor Right Right third floor Right third floor
Load Load/kN /kN
400 400
300 300
403kN 403kN
350kN 350kN
336kN 336kN
200 200
100 100
00
-200 -200
-150 -150
-100 -100
-50 -50
00
50 50
Moment /kN-m /kN-m Moment
100 100
150 150
200 200
Figure 20.20. Load-Moment at the theright rightjoints joints and support. Figure 20. Load-Momentcurves curves at the right joints and support. Figure Load-Moment curves and support.
3.6. Dynamic Dynamic Analysis of the the Two-Span,Six-Story Six-Story Vogel Vogel Steel Steel Frame with Nonlinear Semi-Rigid Connections 3.6. Analysis of Two-Span, Six-Story with Nonlinear Semi-Rigid Connections 4.6. Dynamic Analysis of the Two-Span, Vogel SteelFrame Frame with Nonlinear Semi-Rigid Connections Figure 21 shows the configuration and loading of the two-span, six-story Vogel steel frame.
Figure 21 shows configuration and and loading loading of six-story Vogel steelsteel frame. Figure 21 shows thethe configuration ofthe thetwo-span, two-span, six-story Vogel frame. Each span span and and each each floor floor height height were were 66 m m and and 3.75 3.75 m, m, respectively. respectively. The The section section types types of of beams beams were were Each Each span and each floor height were 6 m and 3.75 m, respectively. The section types of beams IPE240, IPE300, IPE300, IPE360 IPE360 and and IPE400, IPE400, while while the the section section types types of of columns columns were were HEB160, HEB160, HEB200, HEB200, IPE240, were HEB220, IPE240, IPE300, IPE360 and IPE400, while the section types of HEB200, HEB240 and and HEB260. An initial initial geometric imperfection of columns 1/450 was waswere takenHEB160, into account. account. HEB220, HEB240 HEB260. An geometric imperfection of 1/450 taken into HEB220, HEB240 and HEB260. An initial geometric imperfection of 1/450 was taken into account. The distributed distributed loads loads of of 31.7 31.7 kN/m kN/m and and 49.1 49.1 kN/m kN/m applied applied on on the the beams beams were were converted converted into into lumped lumped The The distributed loads ofThe 31.7elastic kN/m and 49.1 kN/m applied onwere the 205 beams converted lumped masses at at the the joints. The elastic modulus and the Poisson Poisson ratio were 205 GPa GPawere and 0.3. 0.3. In order order into to more more masses joints. modulus and the ratio and In to intuitively study the energy consumption of semi-rigid connections, the material of all the frame masses at the joints. The elastic modulus and the Poisson ratio were 205 GPa and 0.3. In order intuitively study the energy consumption of semi-rigid connections, the material of all the frame to members was assumed to be be perfectly perfectly elastic, and and the viscous viscousconnections, damping was wasthe ignored. moremembers intuitively study the energy consumption of semi-rigid material of all the frame was assumed to elastic, the damping ignored. members was assumed to be perfectly elastic, and the viscous damping was ignored.
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In this study, there were 10 contact elements for each column while 16 contact elements for every In this study, there were 10 contact elements for each column while 16 contact elements for every beam. The nonlinear behavior of the semi-rigid connection was simulated by the Richard-Abbott
beam. The nonlinear behavior of the semi-rigid connection was simulated by the Richard-Abbott model. model. The parameters of the model were as follow: Rki = 12336.86 kN ⋅ m / rad ; Rkp = The parameters of the model were as follow: Rki = 12336.86 kN · m/rad; Rkp = 112.97 kN · m/rad; M 0 The m / nrad=; 1.6. = 180kN ⋅ m ; n = 1.6stiffness . The instantaneous stiffness of theatcontact elements at M0 =112 180.97kN kN ·⋅ m; instantaneous of the contact elements semi-rigid connections semi-rigid connections was (13) determined Equations (13) of and The frequency of 1.66 the dynamic was determined by Equations and (15).byThe frequency the(15). dynamic force ω is rad/s, which force ω is 1.66rad / s , which was close to the fundamental natural frequency of this structure. The was close to the fundamental natural frequency of this structure. The dynamic response analysis dynamic response analysis of the frame was performed at three cases of rigid connection, linear of the frame was performed at three cases of rigid connection, linear semi-rigid connection and semi-rigid connection and nonlinear semi-rigid connection, as shown in Figure 22. The comparison nonlinear semi-rigid connection, as shown in Figure 22. The comparison of the numerical results of the numerical results obtained using the modified MDEM and those presented by Chui and Chan obtained using the modified MDEM presented bycan Chui and Chan [13] [13] illustrates that the two match well,and and those the modified MDEM accurately capture theillustrates nonlinear that the two match well, and the modified MDEM can accurately capture the nonlinear behavior behavior of the semi-rigid connection. In addition, it is observed that resonance occurs in the framesof the semi-rigid connection. addition, connections, it is observed occurs in nonlinear the framessemi-rigid with rigid or with rigid or linearInsemi-rigid butthat notresonance in the frame with reason is that the hysteresis damping of the nonlinear connection causes the energy linearconnections, semi-rigid the connections, but not in the frame with nonlinear semi-rigid connections, reason is dissipation (Figure 23). of the nonlinear connection causes energy dissipation (Figure 23). that the hysteresis damping 31.7kN/m
Ψ=1/450 6m
3.75m 3.75m
3.75m
49.1kN/m
49.1kN/m
3.75m
49.1kN/m
3.75m
HEB240
IPE400
HEB260
IPE360
49.1kN/m
49.1kN/m
Ψ
G
Ψ
Lumped mass converted from the vertical distributed load
F1(t)=10.23sin(ωt) kN F2(t)=20.44sin(ωt) kN
3.75m
HEB200 HEB200 HEB240
IPE360
HEB260
IPE300
HEB220
F2(t)
HEB160
F2(t)
HEB160
F2(t)
HEB220
F2(t)
IPE300
HEB220
F2(t)
IPE240
HEB220
F1(t)
6m
Figure Configurationand andloading loading of six-story Vogel steelsteel frame. Figure 21.21. Configuration ofthe thetwo-span, two-span, six-story Vogel frame.
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ω=1.66rad/s ω=1.66rad/s
2
1
Horizontal displacement /m
Horizontal displacement /m
2
1
0
0
-1
-1
-2
Chui and Chan -Rigid Chui andChan Chan-Linear -Rigid semi-rigid Chui and Chui andChan Chan-nonlinear -Linear semi-rigid Chui and semi-rigid
-2
-3
Chui and Chan -nonlinear semi-rigid
-3
0
MDEM
MDEM MDEM MDEM MDEM MDEM
5
0
10
5
10
Time/s/s Time
15
15
20
20
response of the top joint. Figure 22.22. Horizontal response the joint. Figure Horizontaldisplacement displacement response ofof the toptop joint. 120 120
MDEM MDEM Nguyenand andKim Kim Nguyen
80
80 40
40 0 -0.02
-0.02
-0.01
-0.01
00.00 0.00
0.01
0.01
0.02
0.02
-40
-40 -80
-80 -120
Figure 23. Hysteresis loops -120 at the semi-rigid connection G.
In order to simulate the23. collapse behavior ultimate G. strain was taken as the Figure Hysteresis loopsofatthis thestructure, semi-rigid connection Figure 23. semi-rigidthe connection G. fracture criterion. The material was elastic during the calculation, and the beam-column joints were assumed to be linear semi-rigid connections. The sine wave with the frequency of 1.66 rad/s was In order order to to simulate simulate the the collapse collapse behavior behavior of of this this structure, structure, the the ultimate ultimate strain strain was was taken taken as as the the In applied. The failure processes of steel frames with different connections are shown in Figures 24 and 25. fracture criterion. criterion. The The material material was elastic elastic during during the the calculation, calculation, and and the the beam-column joints joints were fracture It can be seen from the figures was that the displacement at the top of the frame withbeam-column semi-rigid connection were assumed to be linear semi-rigid connections. The sine wave with the frequency of 1.66 rad/s was assumed to be linear sine wave with ofof1.66 is obviously largersemi-rigid than that ofconnections. the frame withThe rigid connection, and the the frequency fracture time the rad/s frame was applied. The failure processes of steel frames with different connections are shown in Figures 24 applied. processes of steel frames with different connections are shown in Figures 24 and and 25. 25. withThe rigidfailure connection is earlier. It also indicates that the ductility of the semi-rigid connection is better It can canthan be seen seen from the figures that the displacement at the theconnections top of with semi-rigid connection that of rigidthe connection. Steel frames with semi-rigid have more It be from figures that the displacement at top of the the frame frame withanti-collapse semi-rigid ability connection is obviously obviously larger thanthat thatofof frame with rigid connection, the fracture the frame than those withthan rigid connection. is larger thethe frame with rigid connection, andand the fracture timetime of theofframe with
with rigid connection is earlier. also indicates thatductility the ductility the semi-rigid connection is better rigid connection is earlier. It alsoItindicates that the of theofsemi-rigid connection is better than than that of rigid connection. Steel frames with semi-rigid connections have more anti-collapse ability that of rigid connection. Steel frames with semi-rigid connections have more anti-collapse ability than than those with rigid connection. those with rigid connection.
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(a) 2 s (a) 2 s
(b) 3 s (b) 3 s
(c) 4 s (c) 4 s
Figure 24. Failure Failure processes processes of of the steel steel frame with with rigid connection. connection. Figure 24. Figure 24. Failure processes of the the steel frame frame with rigid rigid connection.
(a) 2 s (a) 2 s
(b) 5 s (b) 5 s
(c) 5.5 s (c) 5.5 s
Figure 25. Failure processes of the steel frame with semi-rigid connection. Figure of the the steel steel frame frame with with semi-rigid semi-rigid connection. connection. Figure 25. 25. Failure Failure processes processes of
5. Conclusions 5. Conclusions 5. Conclusions (1) This paper presented an effective numerical approach for static and dynamic behavior (1) This paperofpresented anwith effective numericalbased approach static Discrete and dynamic behavior steel frames semi-rigid thefor Member Element Method (1) simulation This paper presented an effective numerical joints approach foron static and dynamic behavior simulation of simulation of steel frames with semi-rigid joints based on the Member Discrete Element Method (MDEM). Inwith the MDEM, a structure is on discretized intoDiscrete a set ofElement finite rigid particles, as well as steel frames semi-rigid joints based the Member Method (MDEM). In the (MDEM). Innonlinearity the MDEM,and a structure isbehaviors discretized into a naturally set of finite rigid particles, as well as geometric fracture captured. A virtual spring MDEM, a structure is discretized into a set of finite can rigidbe particles, as well as geometric nonlinearity geometric nonlinearity and fracture behaviors canthe besemi-rigid naturally connection. captured. A virtual spring element without actualcan length is applied to simulate Onactual this basis, and fracture behaviors be naturally captured. A virtual spring element without lengththe is element without actual iselement applied stiffness to simulatethe thesemi-rigid semi-rigidconnection connection.isOn this basis, the modified ofthe thelength contact derived. applied toformula simulate semi-rigid connection. Onatthis basis, the modified formula of the Finally, contact modified formula of the contact element stiffness at the semi-rigid connection is derived. Finally, the numerical approach proposed is verified complex behaviors of steel approach frames with semielement stiffness at the semi-rigid connection is by derived. Finally, the numerical proposed the numerical approach proposed is verified by complex behaviors of steel frames with semirigid connections suchbehaviors as geometric nonlinearity, snap-through buckling, dynamic responses is verified by complex of steel frames with semi-rigid connections such as geometric rigid connections such as geometric nonlinearity, snap-through buckling,taking dynamic responses and fracture. In addition, compared with other numerical approaches the FEM as a nonlinearity, snap-through buckling, dynamic responses and fracture. In addition, compared with and fracture. Inthe addition, compared with other numerical approaches taking the the semi-rigid FEM as a representation, approach proposed is simple and feasible for simulating other numerical approaches taking the FEM as a representation, the approach proposed is simple representation, the approach proposed is element simple and feasible for simulating semi-rigid connections the zero-length spring is because not directly involved in spring thethe calculation. and feasible because for simulating the semi-rigid connections the zero-length element is because theanalysis zero-length spring element is notapproach directly involved the calculation. (2) connections The comparison of the results of the proposed and the in existing researches not directly involved in the calculation. (2) The comparison of the analysis results of the proposed approach and nonlinear the existing researches that the of modified MDEM can accurately capture linear and behaviors of (2) shows The comparison the analysis results of the proposed approach and the existing researches shows shows that the modified MDEM can accurately capture linear and nonlinear behaviors of semi-rigid connections. Some common conclusions canand be nonlinear drown asbehaviors follow: the semi-rigid that the modified MDEM can accurately capture linear of semi-rigid semi-rigid connections. Some reduce common conclusions can and be drown as bearing follow: capacity the semi-rigid connections may significantly structural stiffness, structural under connections. Some common conclusions can be drown as follow: the semi-rigid connections connections may significantly reduce structural stiffness, and structural bearing capacity under static loading will be overestimated if theand semi-rigid are ignored; When the may significantly reduce structural stiffness, structuralconnections bearing capacity under static loading static loading will be load overestimated ifclose the to semi-rigid connections arefrequency, ignored; When the frequency of dynamic applied isconnections structural fundamental will be overestimated if the semi-rigid are ignored; When the frequency ofresonance dynamic frequency of dynamic loadrigid applied linear is close to structural fundamentalnot frequency, resonance occurs in the semi-rigid connections, in the the frames frame with load applied isframes close towith structuralor fundamental frequency, resonancebut occurs in with occurs in the frames with rigid or linear semi-rigid connections, but not in the frame with nonlinear semi-rigid connections, thebut reason that the with hysteresis damping of the nonlinear rigid or linear semi-rigid connections, not inisthe frame nonlinear semi-rigid connections, nonlinear semi-rigid connections, theFracture reason isbehavior that theanalysis hysteresis damping of the nonlinear connection causes energy dissipation. also indicates that frames with the reason is that the hysteresis damping of the nonlinear connection causes energy dissipation. connection causes energy dissipation. Fracture behavior analysis also indicates that frames with semi-rigid connection possess more anti-collapse capacity. Fracture behavior analysis also indicates that frames with semi-rigid connection possess more connection more anti-collapse capacity. (3) semi-rigid The MDEM can avoidpossess the difficulties of finite element method (FEM) in dealing with strong anti-collapse capacity. (3) The MDEM can avoid the difficulties of finite element method (FEM)isinapplied dealing with strong nonlinearity and discontinuity. A unified computational framework static and (3) The MDEM can avoid the difficulties of finite element method (FEM) in dealingfor with strong nonlinearity and discontinuity. A unified computational framework is applied for static and dynamic analyses. The method is simple and generally good, which is an effective tool to nonlinearity and discontinuity. A unified computational framework is applied for static and dynamic analyses. The method is simple and generally good, which is an effective tool to investigate complex The behaviors of issteel frames semi-rigid the follow-up dynamic analyses. method simple andwith generally good,connections. which is an In effective tool to investigate complex behaviors frames with semi-rigid connections. the follow-up study, material nonlinearity willofbesteel taken into account to simulate the collapseInprocess of steel study, nonlinearity will be under taken into account to simulate the collapse process of steel framesmaterial with semi-rigid connections strong earthquake or impact loading action. frames with semi-rigid connections under strong earthquake or impact loading action.
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investigate complex behaviors of steel frames with semi-rigid connections. In the follow-up study, material nonlinearity will be taken into account to simulate the collapse process of steel frames with semi-rigid connections under strong earthquake or impact loading action. Acknowledgments: This work is supported by the National Science Found for Distinguished Young Scholars of China (Grant No. 51125031) and A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (CE02-1-31). Author Contributions: Jihong Ye and Lingling Xu modified the Member Discrete Element Method, analyzed the numerical results, and wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.
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